A Pythagorean Metric in Relativity

Abstract

A one-toone mapping of the class ℝ^1,3 of the timelike vectors of ℝ^1,3 onto ℝ^4,0 is introduced in which i) the pseudo-Pythagorean norm of each element w ∈ ℝ^1,3 and the Pythagorean norm of the corresponding element v ∈ ℝ^4,0 are equal, ii) the “space” parts of w and v are proportional, and iii) the transformation properties of v are induced by those of w. With the aid of such a mapping, a new interpretation of the rest energy of a particle of special relativity is proposed. Half of this quantity is the sum of two terms, one of which formally coincides with the classical kinetic energy of a point like particle although it involves the relativistic velocity instead of the classical one, so it is called the quasi-classical kinetic energy. The other one is interpreted as describing an internal degree of freedom of the particle. As a result, the rest energy, which is invariant (i.e., independent of frame), is regarded as twice the amount of the total kinetic energy of the particle. This accounts for the acronym itke.